MEN OF MATHEMATICS
delivered their verdict that my "faithful pupil", my "weak-
ness" is indeed not a 4'frivolous humbug".'
We may leave the friends in their moment of triumph. TVo
years later (10 February 1891) Sonja died in Stockholm at the
age of forty-one after a brief attack of influenza which at the
time was epidemic. Weierstrass outlived her six years, dying
peacefully in his eighty-second year on 19 February 1897 at his
home in Berlin after a long illness followed by influenza. His
last wish was that the priest say nothing in his praise at the
funeral but restrict the services to the customary prayers.
Sonja is buried in Stockholm, Weierstrass with his two sisters
in a Catholic cemetery in Berlin. Sonja also was of the Catholic
faith, belonging to the Eastern Church.
We shall now give some intimation of two of the basic ideas
on which Weierstrass founded his work in analysis. Details or an
exact description are out of the question here, but may be found
in the earlier chapters of any competently written book on the
theory of functions.
A power series is an expression of the form
00 -r ^2 -r a^ ->-...+ anzn +-...,
in which the coefficients a0, a19 a2 ..., an> ... are constant
numbers and z is a variable number; the numbers concerned
may be real or complex.
The sums of 1,2,3, ... terms of the series, namely a0, aQ -f
&& ac -f o^z -f ci?z2, ... are called the partial sums. If for some
particular value of z these partial sums give a sequence of
numbers which converge to a definite limit, the power series is
said to converge to the same limit for that value of 2.
All the values of z for which the power series converges to a
limit constitute the domain of convergence of the series; for any
value of the variable z in this domain the series converges; for
other values of z it diverges.
If the series converges for some value of z9 its value can be
calculated to any desired degree of approximation, for that
value, by taking a sufficiently large number of terms.
Now, in the majority of mathematical problems which have
applications to science, the 'answer' is indicated as the solution